The banker's gain on a certain sum due 2 years hence at 10% per annum is Rs 24. What is the present worth of this sum (in rupees)?

Introduction / Context:
This question belongs to the topic of banker's discount, true discount, banker's gain and present worth. The banker's gain is the difference between the banker's discount and the true discount. By using standard formulas for a given rate and time, we can relate banker's gain to the face value and present worth of the bill. Here we are told the banker's gain and asked to find the present worth at a simple interest rate of 10 percent per annum for 2 years.

Given Data / Assumptions:
- Time until the sum is due = 2 years
- Rate of simple interest = 10 percent per annum
- Banker's gain (BG) = Rs 24
- Let face value of the bill be F
- Present worth is the true value of the sum today at the given rate and time

Concept / Approach:
For a bill due after time t years at rate r percent per annum, banker's discount BD is calculated on the face value F as BD = F × r × t / 100. True discount TD is the difference between the face value and present worth. A useful relation is that banker's gain BG equals BD minus TD and for given r and t can be expressed as a fixed fraction of the face value. Once we find F from BG, we can compute present worth using the true discount relations.

Step-by-Step Solution:
Step 1: For r = 10 percent and t = 2 years, we have r × t = 20.Step 2: True discount TD = F × 20 / (100 + 20) = F × 20 / 120 = F / 6.Step 3: Banker's discount BD = F × 20 / 100 = F / 5.Step 4: Banker's gain BG = BD − TD = F / 5 − F / 6 = F / 30.Step 5: Given BG = 24, so F / 30 = 24 which implies F = 24 × 30 = Rs 720.Step 6: True discount TD = F / 6 = 720 ÷ 6 = Rs 120.Step 7: Present worth = face value − true discount = 720 − 120 = Rs 600.

Verification / Alternative check:
We can verify the result by recomputing BD from F. With F = 720, rate 10 percent and time 2 years, BD = 720 × 10 × 2 / 100 = 720 × 20 / 100 = Rs 144. Since TD = Rs 120, the banker's gain is BD − TD = 144 − 120 = Rs 24, which matches the given gain. Therefore the computed present worth of Rs 600 is consistent with all definitions and data.

Why Other Options Are Wrong:
A present worth of Rs 400 or Rs 500 would lead to a face value and discounts that do not produce a banker's gain of Rs 24 at 10 percent for 2 years. A value like Rs 800 is larger than the face value we found and is impossible for present worth. The face value itself is Rs 720, so present worth must be smaller than this. Only Rs 600 satisfies both the algebraic relations and the given banker's gain.

Common Pitfalls:
Students sometimes confuse banker's gain with banker's discount or take BG as a percentage instead of an amount. Others use BD directly to find present worth, ignoring the role of true discount. It is important to remember that BG = BD − TD and that true discount is linked to present worth, not directly to the face value in the same way as banker's discount.

Final Answer:
The present worth of the sum is 600 rupees.